To determine the density of H₂O₂ from a measured absorption spectrum, a fitting procedure similar to that reported in references [10, 36] has been employed. The absorption spectrum was obtained by measuring the ring-down times τ(ν) and τ ₀ at different spectral positions, while the plasma jet was switched on and off, respectively, according to: 1 c τ ν − 1 c τ 0 = α ν d L . (3) Here, α(ν) is the absorption coefficient, L is the cavity length, d is the diameter of the volume, in which the absorbing species are located (i.e., the effective absorption length), and c is the speed of light in vacuum. The frequency dependence of τ ₀ within the applied tuning range of the QCL was small. Hence, τ ₀ was considered to be constant over the recorded spectral range of a spectrum.

The measured absorption α ( ν ) d L is a result of the contributions from the absorption of H₂O₂ distributed with a density [H ₂ O ₂] within a volume with a diameter of d H 2 O 2 , and from (broadband) absorptions of other molecules, which were represented by a straight line with the coefficients b ₀ (y-intercept) and b ₁ (slope): α ν d L = b 0 + b 1 ν + d H 2 O 2 L H 2 O 2 σ H 2 O 2 ν + ϵ ν . (4) Here, σ H 2 O 2 ( ν ) is the frequency dependent absorption cross section of H₂O₂, and ϵ(ν) is the residual of the fitting procedure. The frequency dependent absorption cross section was computed by using a Voigt function V ν − ν 0 , t ; γ H 2 O 2 with different values for the pressure broadening coefficient γ H 2 O 2 according to: σ H 2 O 2 ν ; γ H 2 O 2 = ∑ t S t V ν − ν 0 , t ; γ H 2 O 2 . (5) Here, γ H 2 O 2 denotes the pressure broadening coefficient for H₂O₂ chosen as a constant value, while the considered transitions t of H₂O₂ at the frequencies ν _0,t , and the corresponding line strengths S _t were taken from the HITRAN database [30–32, 43]. In the effluent of the kINPen-sci plasma jet, a mixture of H₂O, O₂, air, and Ar is expected, which results in a significantly different pressure broadening coefficient than the one reported in the HITRAN database. The pressure broadening coefficient was assumed to be the same for all transitions, as it was reported in the HITRAN database for the pressure broadening coefficients in air [30–32, 43], and previously done for the pressure broadening coefficients for HO₂ [10].

For the determination of [H ₂ O ₂], and of γ H 2 O 2 , the sum of squares of the residuals χ 2 = ∑ j = 1 N ν ϵ 2 ( ν j ) in Eq. 4 at the measured spectral positions ν _j were minimised for different values for γ H 2 O 2 , such that ∂ χ 2 ∂ β = 0 , which is equivalent to solve a matrix-equation with the form M β = ω. Here, β = b 0 , b 1 , [ H 2 O 2 ] includes the fitting parameters, M is a 3 × 3-matrix: M = N ν ∑ j = 1 N ν ν j ∑ j = 1 N ν σ H 2 O 2 ν j ∑ j = 1 N ν ν j ∑ j = 1 N ν ν j 2 ∑ j = 1 N ν σ H 2 O 2 ν j ν j ∑ j = 1 N ν σ H 2 O 2 ν j ∑ j = 1 N ν σ H 2 O 2 ν j ν j ∑ j = 1 N ν σ H 2 O 2 ν j 2 , (6) where N _ν is the total number of the measured spectral positions, and ω comprises the experimental data and the calculated frequency dependent absorption cross sections σ H 2 O 2 ( ν j ) with the chosen values for γ H 2 O 2 : ω = 1 c ∑ j = 1 N ν τ ν − 1 − τ 0 − 1 ∑ j = 1 N ν τ − 1 − τ 0 − 1 ν j ∑ j = 1 N ν τ − 1 − τ 0 − 1 σ H 2 O 2 ν j . (7) The values for β that yielded the lowest of squares of the residuals have been retained as the best fit parameters. The error for [H ₂ O ₂] and for γ H 2 O 2 was evaluated by using the same F-ratio method as described in reference [10]. As the procedure to obtain a full spectrum at various radial and axial positions is time consuming, in this work, also the on/off-resonance method has been employed presented by [36] for the measurements of HO₂, in particular to determine the effective absorption length, which will be described in the following.

Based on the on/off-resonance method, radial and axial scans through the effluent have been performed by recording 100 ring-down events at ν _on = 1,231.07 cm⁻¹ for the on-resonance position, and at ν _off = 1,230.82 cm⁻¹ for the off-resonance position. The broadband absorptions of HNO₃, HONO, and N₂O₅ were assumed to be the same at the positions for the on- and the off-resonance such that Δα is defined by: Δ α ≔ 1 c τ ν o n − 1 c τ ν off = d H 2 O 2 L H 2 O 2 Δ σ H 2 O 2 . (8) Here, c is the speed of light in vacuum, τ(ν _on ) and τ(ν _off ) are the measured ring-down times at the on-resonance and off-resonance position, respectively, while the plasma jet was switched on, d H 2 O 2 is the absorption length for H₂O₂, [H ₂ O ₂] is the density of H₂O₂ located in a volume with a diameter of d H 2 O 2 along the line-of-sight of the laser beam, and Δ σ H 2 O 2 ≔ σ H 2 O 2 ( ν o n ) − σ H 2 O 2 ( ν off ) is the difference of the frequency dependent absorption cross sections at the on-resonance and the off-resonance position. The assumption for a constant value for the broadband absorbers is valid, as long as the slope for the baseline in the fitting procedure is small compared to the error of the measured absorption.

Several full spectra at various axial distances z from the nozzle were recorded, in order to validate the on/off-resonance method, and to determine the value for the pressure broadening coefficient, γ H 2 O 2 , which is crucial for the application of the on/off-resonance method. In Figure 4, absorption spectra for various z-distances between 3 mm and 10 mm below the nozzle of the kINPen-sci plasma jet are depicted, together with a fit of the baseline and the contribution of the absorption of H₂O₂. For each spectrum, an absorption feature with a shape similar to the H₂O₂ absorption cross sections as reported by the HITRAN database was observed at each z-position, while a pressure broadening coefficient was determined from the best fit as described in Section 3.1. Therefore, 221 transitions for H₂O₂ with line strengths ranging from approximately 10^–25 cm²·cm⁻¹ to approximately 10^–20 cm²·cm⁻¹ were considered. In Figure 5, the frequency dependent absorption cross section σ H 2 O 2 ν ; γ H 2 O 2 for H₂O₂ at 300 K with a pressure broadening coefficient of γ H 2 O 2 = 0.06 cm⁻¹atm⁻¹ is depicted as calculated according to Eq. 5, together with the corresponding line strengths S _t of the considered 221 H₂O₂-transitions, which were taken from the HITRAN database [30–32, 43]. The peak height of the spectra shown in Figure 4 was varying due to a subjacent baseline, whose mean value increased with increasing z-position. This baseline accounts for broadband absorptions resulting from transitions of HNO₃, HONO, and N₂O₅, which could not be further specified due to their broadband character. In Figure 6, the corresponding pressure broadening coefficients γ H 2 O 2 for the observed H₂O₂ transitions determined from the spectra shown in Figure 4 are depicted as a function of the distance from the nozzle z. Within the error, no significant relation between the z-position of the kINPen-sci plasma jet, and the pressure broadening coefficient for H₂O₂ could be observed. This indicates that, within the margin of error, a change of the gas mixture at various z-positions due to the diffusion of the gas curtain into the effluent was not influencing the pressure broadening coefficient of the investigated transitions of H₂O₂. Hence, a mean value of γ H 2 O 2 = 0.06 cm⁻¹atm⁻¹ was chosen for all transitions, and for all radial and axial positions. Notably, this value is by a factor of 0.6 smaller than the value for the pressure broadening coefficient in air reported in the HITRAN database [30–32, 43]. A similar relation between the pressure broadening coefficients in air and the one determined for the gas mixture in the kINPen-sci plasma jet has been found previously for HO₂ reported by [10]. For the following analysis, Δσ was determined by using the mean value of the frequency dependent absorption cross section to be: Δ σ H 2 O 2 = 1.6 ⋅ 1 0 − 19 c m 2 . (9)

As it has been shown that broadband absorptions from other species than H₂O₂ play a significant role in the absorption spectra, the absorption of H₂O₂ has been compared qualitatively to the broadband absorptions. Therefore, the evolution of the baseline was correlated to the evolution of the absorption of H₂O₂, both as a function of z. For all spectra, the slope of the baseline was small compared to the error of the measured absorption. Hence, considering a single wavelength position is sufficient for the following comparison of the absorption of H₂O₂ and the baseline. In order to facilitate this comparison, which includes information on the shape of the absorption cross section for H₂O₂ as known so far, the absorption for H₂O₂ was evaluated by assuming a constant absorption length of d = 4 mm and a line-of-sight integrated density for H₂O₂ was determined. In Figure 7, the line-of-sight integrated density of H₂O₂ is depicted as a function of the distance from the nozzle z, together with the value of the baseline at 1,231.07 cm⁻¹. The density of H₂O₂ increased between z = 3 mm and z = 8 mm linearly by approximately a factor of 2.3 to its maximum 9.7 ⋅ 10¹³ cm⁻³, and decreased to 7.1 ⋅ 10¹³ cm⁻³ between z = 6 mm and z = 10 mm. Within the error of the density, the baseline had a similar trend; the baseline increased up to z = 8 mm, and decreased between z = 8 mm and z = 10 mm. This indicates that besides H₂O₂, also the densities of HNO₃, HONO, and N₂O₅ increased. However, the effective absorption length needs to be considered to compare the densities at various z-positions. Due to the on/off-resonance method employed in this work, this can only be performed for the density of H₂O₂.

In order to obtain the effective absorption length, radial scans at various z-positions between z = 3 mm and z = 10 mm have been recorded by using the on/off-resonance method. Due to the diameter of the observed cavity mode, which was 5.5 mm in the focal point of the cavity (containing 99% of the intensity), no ring-down measurement closer than 3 mm below the nozzle could be performed. In Figure 8, radial scans for Δα as a function of the distance to the symmetry axis through the kINPen-sci plasma jet, x, are depicted for z = 3, 4, 6, 8, and 10 mm together with a Gaussian fit, and a line representing the effective absorption length. For all z-positions, a Gaussian distribution for Δα was determined.

In Figure 9, the centre positions, x _c , for the Gaussian fits of the radial scans are illustrated as a function of the distance from the nozzle z. Here, x _c = 0 mm denotes the position of the symmetry axis of the kINPen-sci plasma jet in the centre of the nozzle. For the measured z-positions, the centre of the Gaussian fits were distributed between −0.4 and 0.4 mm relative to the centre of the nozzle. However, the weighted mean of the centre positions was at −0.12 mm, which indicates a slight asymmetry in the H₂O₂ distribution. A similar asymmetry has been reported previously for HO₂ [36] and for O and H atoms in the same plasma jet [36]. Within the plasma zone of the plasma jet, the discharge is likely to be asymmetric because of inhomogeneities of the powered needle electrode, for instance, which results in an asymmetric production of reactive species.

The effective absorption length d was determined by the limits that contain 99% of the area A(z) of the Gaussian fit: d z = 2.576 ⋅ w z . (10) Here, w(z) is the width of the Gaussian function, which is defined by: Δ α x , z = A z w z π 2 exp − 2 x − x c w z 2 + Δ α off . (11) Here, x _c is the centre position of the Gaussian fit, and Δα _off is an offset value.

Although the closest measurement could be taken at 3 mm from the nozzle, the localised density distribution of H₂O₂ close to the nozzle can be obtained by extrapolating the absorption lengths to z = 0 mm from the nozzle. Therefore, the effective absorption lengths determined for H₂O₂ have been compared to the absorption lengths d H O 2 determined previously for HO₂ [36], and to the region with the diameters d _H , d _O , where O and H atoms were distributed within the effluent, respectively, as reported in reference [47]. This is a reasonable approach, since reactive species, including H₂O₂, HO₂, H atoms, and O atoms, generated in the plasma zone and in the effluent. Consequently, H₂O₂ is expected to be present in the same region as other reactive species that are precursors for H₂O₂. Furthermore, it is known from previous investigations that the gas curtain starts to diffuse into the effluent efficiently from 4 mm below the nozzle [48], [35], which introduces further species besides species that already have been present in the plasma zone, and which can result into a further production of H₂O₂. Hence, the region, where H₂O₂ is present close to the nozzle before the gas curtain starts to diffuse into the effluent efficiently, can be expected to be the same as for other reactive species generated within the plasma zone. In Figure 10, the determined values for d _H /2, d _O /2, d H O 2 / 2 , and d H 2 O 2 / 2 for the species distributions of H, O, HO₂, and H₂O₂ are illustrated, as they are related to the radius of the nozzle of the plasma jet to be half of the absorption lengths, as a function of the distance to the nozzle z together with a polynomial approximation described by: d H O 2 , H 2 O 2 / 2 = 1.6 − 0.1 ⋅ z + 0.33 ⋅ z 2 − 0.057 ⋅ z 3 + 0.0039 ⋅ z 4 − 9.4 ⋅ 1 0 − 5 ⋅ z 5 . (12) By using this polynomial approximation, information close to the nozzle can be extrapolated, which was not possible to obtain experimentally due to the inference of the plasma jet with the laser beam in the cavity.

Based on the determined effective absorption length d H 2 O 2 presented in Eq. 12, line-of-sight integrated densities have been obtained as a function from the distance to the nozzle z from full spectra and from an axial scan Δα(z) recorded by using the on/off-resonance method. In Figure 11, the line-of-sight integrated density distributions of H₂O₂ in the effluent of the kINPen-sci plasma jet are depicted as a function of the distance from the nozzle z. Within the error, the densities obtained from both methods are in agreement with each other. Between z = 3 mm and z = 6 mm, the density of H₂O₂ plateaued at approximately 7.5 ⋅ 10¹³ cm⁻³ within the error, and decreased linearly for further distances from the nozzle to approximately 5 ⋅ 10¹³ cm⁻³ at z = 10 mm.

One drawback of the employment of absorption spectroscopy is, however, that only line-of-sight integrated densities are obtained. In order to transform the line-of-sight integrated absorption into a radial density distribution, an Abel inversion was employed, since the kINPen-sci plasma jet can be approximated by a cylindrical symmetry. By applying an Abel inversion on the radial scans Δα(x, z) presented in Figure 8 according to: Δ α ρ = − 1 π ∫ ρ R d Δ α x , z d x d x x − x c 2 − ρ 2 , (13) the radial density distribution was obtained. Here, ρ is the radial dimension described by ρ = ( x − x c ) 2 + y 2 , x _c is the centre position of the radial distributions yielding the distance to the symmetry axis through the nozzle of the plasma jet, and R is the effective absorption length divided by 2. As all radial scans could be represented by a Gaussian function, the analytical solution for the Abel-transformed absorption coefficient Δα(ρ) equals: Δ α ρ = 2 π A z w 2 z exp − 2 ρ 2 w 2 z . (14) Here, A(z) and w(z) are the area and the width of the Gaussian distributions for the radial scans, respectively. The radial density distribution n H 2 O 2 ( ρ ) of H₂O₂ was then determined by: n H 2 O 2 ρ = Δ α ρ L Δ σ H 2 O 2 . (15) Here, L is the cavity length, and Δ σ H 2 O 2 is the absorption cross section determined for the on/off-resonance method presented in Eq. 9.

In Figure 12, the radial density distributions for H₂O₂ are illustrated for various z-positions. Except for the measurement at z = 4 mm, the maximum intensities decreased with further distance from the nozzle. However, for the measurement at z = 4 mm, a maximum density of twice the value for the maximum density at z = 6 mm and a factor of approximately 1.5 more than the value for the maximum density at z = 3 mm was obtained. In comparison to the measurements of full spectra and the axial scan along the symmetry axis of the plasma jet at x = 0 recorded by the on/off-resonance method, the value for Δα obtained from the radial scan at z = 4 mm is the only value that does not agree with the previous measurements within the error. Furthermore, at z = 4 mm the maximum was shifted by approximately x = 0.4 mm compared to the symmetry axis through the plasma jet, which is opposite to most of the centre positions for the radial scans at other z positions as illustrated in Figure 9. Hence, the absolute values for the radial distribution obtained from the radial scan at z = 4 mm is regarded as an outlier. Notably, the radial distribution for H₂O₂ depends strongly on the data quality for the radial scans. Since the measured absorption is greater by a factor of 3 than the scatter of the data points, larger errors are to be expected, especially when determining the center position.

The localised density distribution in the radial and axial dimension was determined from an axial and several radial scans measuring Δα as a function of z and x, respectively, by employing the on/off-resonance method. According to Eq. 15, the localised density distribution for H₂O₂ is given by: n H 2 O 2 x , y , z = L w z π 2 Δ α x = 0 , z Δ σ H 2 O 2 exp − 2 ρ 2 − x c 2 w 2 z . (16) In order to obtain a continuous density distribution, an interpolation for the evolution of the width w(z) determined from several radial scans as described by Eqs 10, 12, and for Δα(x = 0, z) was used. Therefore, Δα(x = 0, z) was described by a polynomial approximation: Δ α H 2 O 2 x = 0 , z = 4.3 ⋅ 1 0 − 8 − 8.9 ⋅ 1 0 − 9 ⋅ z + 1.5 ⋅ 1 0 − 9 ⋅ z 2 − 3.5 ⋅ 1 0 − 9 ⋅ z 3 + 3.5 ⋅ 1 0 − 10 ⋅ z 4 − 1.5 ⋅ 1 0 − 11 ⋅ z 5 + 2.3 ⋅ 1 0 − 13 ⋅ z 6 . (17) Due to the lack of data points close to the nozzle for Δα for H₂O₂ within the first 3 mm below the nozzle, the polynomial approximation in this region was chosen in such a way that the axial density distribution for H₂O₂, which was determined by using the absorption length d H 2 O 2 , was constant within the first 3 mm. This assumption is based on the observation that previously investigated reactive species are mainly generated within the plasma zone of the plasma jet and their density remains approximately constant over the first 3 mm, as it was determined for HO₂ [36]. In Figure 13, Δα(x = 0, z) measured with the on/off-resonance method, is depicted as a function of z together with a polynomial approximation for Δα.

The resulting localised density distribution of H₂O₂ is illustrated as a contour plot in a plane cut along the symmetry axis through the centre of the nozzle in Figure 14A. Lateral profiles of this contour plot at different z-positions are presented in Figure 14B, respectively. The densities of H₂O₂ were mainly distributed within a cone with a diameter of approximately 1.6 mm at z = 0 mm and 5 mm at z = 10 mm. The maximum density of approximately 2 ⋅ 10¹⁴ cm⁻³ H₂O₂ was obtained in the centre of the effluent between z = 0 mm and z = 4 mm. For larger z-distances, the density of H₂O₂ decreased to approximately 1 ⋅ 10¹⁴ cm⁻³ at 6 mm and remained approximately constant thereafter. From the presented localised density distribution, it can be concluded that H₂O₂ is significantly generated within the plasma zone of the plasma jet, as the maximum density was obtained close to the nozzle. The decrease of the density of H₂O₂ between z = 4 mm and z = 6 mm is most likely due to an impact of the surrounding gas composition; a larger amount of O₂ from the gas curtain leads to a net consumption of H₂O₂. A similar impact of the surrounding gas composition has also been reported for O atoms, H atoms [47], and HO₂ radicals [36]. With the strong reduction of species, that consume H₂O₂, such as OH or HO₂ radicals, the density of H₂O₂ is expected to remain constant at distances further from the nozzle, which is in agreement with the measured densities at distances further than 6 mm from the nozzle.